$ = \left( {a + b + c} \right)\begin{array}{*{20}{c}}
1&1&1\\
b&c&a\\
c&a&b
\end{array}$

             $ = \left( {a + b + c} \right)\begin{array}{*{20}{c}}
0&0&0\\
{b - c}&{c - a}&a\\
{c - a}&{a - b}&b
\end{array}$

$ = \left( {a + b + c} \right)\left[ {ab + bc + ca - {a^2} - {b^2} - {c^2}} \right]$

$ =  - \left( {a + b + c} \right)\left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right]$

Since $a,b,c$  are sides of $a$ scalene triangle, therfore at least two of the $a,b,c,$ will be unqual.

$\therefore {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2} > 0$ 

Also $a + b + c > 0$

$\therefore  - \left( {a + b + c} \right)\left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right] < 0$